Physical Foundations of Cosmology

4.5 Instantons, sphalerons and the early universe 189

inside the wall by a factorl/RI1 compared to(dφ/dr ̃)^2 andV.Therefore, in
the leading approximation we have

(dφ/dr ̃)^2 ≈ 2 V

forRI+l>r ̃>RI−l. Using this result, (4.165) simplifies to

−V(φr ̃= 0 )≈

3 σ

RI

, (4.166)

where

σ≡

∫∞

0

(

dφ

dr ̃′

) 2

dr ̃′≈

∫φ ̃r=^0

0

√

2 Vdφ (4.167)

is the surface tension of the bubble. The instanton action (4.159) then becomes

SI≈

π^2

2

V(φr ̃= 0 )R^4 I+ 2 π^2 σR^3 I≈

27 π^2 σ^4

2 |V(φ ̃r= 0 )|^3

, (4.168)

where the first and second terms are the contributions of the internal region of the
bubble and its wall respectively. For the potentialVshown in Figure 4.13 the action
takes the minimal value for|V(φr ̃= 0 )|=.In this case the field inside the bubble is
equal toφ 0 and, as follows from (4.166), the instanton has sizeRI≈ 3 σ/. Hence
the vacuum decay rate is equal to

Aexp

(

−

27 π^2 σ^4

2 ^3

)

. (4.169)

At a “given moment of Euclidean time”τthe solutionφ

(√

r^2 +τ^2

)

describes

a three-dimensional bubble. The half of the instanton connecting the metastable
vacuum with the classically allowed region “evolves” in Euclidean time as follows.
Atτ→−∞we haveφ=0 everywhere in space. “Later on” atτ∼−RIthe bubble

“appears” and its radius “grows” asR(τ)≈

√

R^2 I−τ^2 reaching the maximal value

RIatτ= 0.

Problem 4.24Calculate the potential energy of this bubble and verify that

V(R(τ))≈

2 π

3

R(τ)

(

R^2 I−R^2 (τ)

)

. (4.170)

The total energy corresponding to the instanton solution is equal to zero. There-
fore, a bubble with radius 0<R(τ)<RIis in the classically forbidden region (un-
der the barrier), whereV(R)>0. A three-dimensional bubble of sizeRI≈ 3 σ/
is on the border of the classically allowed region (V(RI)=0) and it “materializes”